Akshay Shankar
2022-10-15
\[\begin{aligned} \hat{H}=& \int d^{3} \mathbf{r} \cdot \psi^{\dagger}(\mathbf{r})\left[-\frac{\hbar^{2} \nabla^{2}}{2 m}+V_{\text {ext. }}(\mathbf{r})\right] \psi(\mathbf{r}) +\frac{1}{2} \int \psi^{\dagger}(\mathbf{r}) \psi^{\dagger}\left(\mathbf{r}^{\prime}\right) \underbrace{V\left(\mathbf{r}^{\prime}-\mathbf{r}\right)}_{V_c + V_d} \psi\left(\mathbf{r}^{\prime}\right) \psi(\mathbf{r}) \cdot d^{3} \mathbf{r} d^{3} \mathbf{r}^{\prime} \end{aligned}\]
\(\hspace{9.5cm}\)nearest-neighbour \(\hspace{1cm}\Bigg\Downarrow \hspace{1cm}\psi(r) = \sum_i w_{R_i}(r) \cdot a_i\)
\[\boxed{H_{BH} = -t\cdot \sum_{\langle
i,j \rangle} a_i^{\dagger}a_j + \frac{U}{2} \sum_i n_i(n_i -1) + V
\sum_{\langle i, j \rangle} n_i n_j}\]
\[\small t_{i, j} = \int dr \cdot w^*_{R_i}(r)
\cdot \hat{H} \cdot w_{R_j}(r)\]
\[\small V_c(r) = \frac{4Ï€\hbar^2 a_s}{m}\cdot\delta(r) = g\cdot\delta(r) U_i \hspace{1cm}\longrightarrow \hspace{1cm} U_i = g \int dr \cdot |w_{R_i}(r)|^4\]
\[\small V_d(r, r') = \frac{\mu_0 \mu_m^2}{4\pi} \cdot \frac{(1 - 3\cos^2\theta)}{(r - r')^3} \hspace{1cm}\longrightarrow \hspace{1cm} V_{i, j} = \int dr^3 dr'^3 \cdot|w_{R_i}(r')|^2 \cdot V_d(r - r') \cdot |w_{R_j}(r)|^2 \]
\[\small H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j\hspace{0.5cm}\longrightarrow\hspace{0.5cm}|\Psi_{SF}\rangle= \frac{1}{N!} (\sum_{i=1}^M a_i^{\dagger})^N |{0}\rangle \hspace{0.5cm}\]
Consider N bosons in M lattice sites.
\[\small \underbrace{H =
-t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i
n_i(n_i - 1)}_{\text{coupled lattice sites}}
\hspace{0.5cm}\longrightarrow\hspace{0.5cm} \underbrace{H \{\Psi \} =
\sum_i-zt \cdot (\Psi^*a_i + \Psi a_i^{\dagger} - |\Psi|^2) +
\frac{U}{2}n_i(n_i -1)}_{\text{de-coupled lattice sites}}\]
\[\small \underbrace{H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1)}_{\text{coupled lattice sites}} \hspace{0.5cm}\longrightarrow\hspace{0.5cm} \underbrace{H \{\Psi_i \} = \sum_C H_{exact} + \sum_{C, C'}H_{MFT}\{ \Psi_i \}}_{\text{de-coupled clusters of sites}}\]